Question

求由曲线Y=1/x,y=x与x=2所围成图形的面积,以及该图形绕x轴旋转而成的立体的体积

主要是第二问..

Answer 1

y = 1/x与交于A(1, 1), 与x = 2交于(2, 1/2)
积分区间为[1, 2]，此时y =x在y = 1/x上方
S = ∫₁²(x - 1/x)dx = (x²/2 - lnx)|₁² = (2 - ln2) - (1/2 - 0) = 3/2 - ln2
V = ∫₁²π(x² - 1/x²)dx = π(x³/3 + 1/x)|₁² = π(8/3 + 1/2) - π(1/3 + 1) = 11π/6

Answer 2

s=∫（0,1）xdx+∫﹙1,2﹚1/x dx=1/2＋ln2
v=∫﹙0,1﹚πx²dx + ∫﹙1,2﹚ π﹙1/x﹚² dx=π1/3 + π1/2=π5/6

Answer 3